Multivariate normal approximation for traces of random unitary matrices

In this article we obtain a superexponential rate of convergence in total variation between the traces first m powers n×n random unitary matrices and 2m-dimensional Gaussian variable. This generalizes previous results scalar case to multivariate setting, also give precise dependence on dimensions n estimates with explicit constants. We are especially interested regime where grows our main result basically states that if m?n, then approximation is ?(nm+1)?1 times correction. show remains valid for all m?n2/3 without fast convergence.